Download File - dr. carruthers @ LCHS

La Cañada Math II Advanced Newsletter
Unit 1 – Geometric Foundations and Reasoning
Unit 1 Overview
In this introductory unit, students will be exposed to the language
that forms the foundation of geometry, such as point, line, plane,
segment, ray, distance, angle, etc. The unit will open with an
introduction to several important undefined terms; point, line and
plane, and how those terms are illustrated geometrically and with
appropriate notation. Students will use Geometer’s Sketchpad to
solidify their understanding of these terms and familiarize
themselves with the protocols for using geometric software. As an
additional representation, points and line segments will be
explored on the coordinate plane, and formulas will be developed
to measure the distance between two points and the midpoint of
the segment joining two points.
Students will then be introduced to the definition of an angle, and
how it can be defined using rays with a common endpoint or in
terms of the arc length of a circle. The definitions discussed in the
first part of this unit will lead to several postulates that students will
use to describe the characteristics of a diagram. An additional
emphasis will be on construction, which is the ability to create a
precise diagram with a straightedge and compass, and no need
for rulers, protractors, or the approximation that they carry with
them. Students will relate each construction to the definition of the
term and use them throughout the course as a connection to
axiomatic logic and proof.
Once the appropriate definitions are in place, the focus of the unit
will shift to the use of inductive and deductive reasoning in
mathematical sense-making. Students will learn how inductive
reasoning allows for conjectures to be formed out of pattern
recognition and that such a conjecture can only be considered
valid if it is proven deductively. Students will use properties of
equality and congruence, along with previous and new definitions
and postulated to prove conjectures. These proofs will require an
understanding of an organizational system (a 2-column proof at
this introductory stage) and logic, in order to prove the assumption
laid out in the conjecture. Proofs will include the Midpoint
Theorem, Angle Bisector Theorem, definition of vertical angles, and
other conjectures involving complementary and supplementary
angles.
Important Dates
August 31
Unit 1 Test (Tentative)
Homework Policy
Daily Homework will be assigned
and should be completed prior to
the next class meeting. For more
details about assignments, check
the class website.
Weekly Homework will be a
component of the course, and will
emphasize algebra skills learned in
last year’s course in an attempt to
maintain important ideas before
entering LC Math III.
Technology Note
We will be using various geometry
software programs this year in class.
Geometer’s Sketchpad is a
program available online, but only
in a trial capacity. In school, we will
visit the Mac Lab to use this
program. Geogebra is a free online
program that will also be useful at
home and in class.
If you are absent from a class
period in which these programs are
used, feel free to reach out to the
teacher to insure that the
assignment can be completed on
a school computer or at home
using the appropriate technology.
Page 1 of 2
Additional Unit 1 Information
Key Common Core Standards
G.CO.1 - Know precise definitions of angle, circle, perpendicular
line, parallel line, and line segment, based on the undefined
notions of point, line, distance along a line, and distance around
a circular arc.
G.CO.9 - Prove theorems about lines and angles. Theorems
include: vertical angles are congruent; when a transversal crosses
parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a perpendicular
bisector of a line segment are exactly those equidistant from the
segment’s endpoints.
G.CO.12 - Make formal geometric constructions with a variety of
tools and methods (compass and straightedge, string, reflective
devices, paper folding, dynamic geometric software, etc.).
Copying a segment; copying an angle; bisecting a segment;
bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line
parallel to a given line through a point not on the line.
8.G.8b - Apply the Pythagorean Theorem to find the distance
between two points in a coordinate system.
G.GPE.6 - Find the point on a directed line segment between two
given points that partitions the segment in a given ratio.
Helpful Resources
 Geometry Textbook – Chapters 1 and 2,
Chapter 13 (Sections 1 and 5 only)
 Construction Support:
www.mathsisfun.com/geometry/constructions.html

Geogebra: www.geogebra.org
Dr. Carruthers ([email protected])
Mr. McDermott ([email protected])
Spotlight on the
Standards for
Mathematical
Practice:
MP 3: Construct viable
arguments and critique the
reasoning of others.
Constructing arguments is an
important mathematical
practice. Mathematics, as a
field of study, is characterized
by a need to prove what is true.
As students advance in their
mathematics education, they
will be asked to justify and prove
mathematical relationships with
an increasing level of rigor and
formality.
This unit will introduce students
to the idea of proof, and
making a logical argument that
is supported by definitions and
postulated. Deductive
reasoning will be a skill that will
take time to develop
throughout the course, but is a
critical byproduct of the
geometry knowledge gained
through this course.
Students will practice
sequencing statements logically
and supporting claims with
evidence, both important skills
that are transferrable to other
disciplines.
Page 2 of 2